On the mod 2 cohomology algebra of oriented Grassmannians

For \(n\in \{2^t-3,2^t-2,2^t-1\}\) $n\in \{2^t-3,2^t-2,2^t-1\}$ \((t\ge 3)\) $(t\ge 3)$ we study the cohomology algebra \(H^*(\widetilde{G}_{n,3};{\mathbb {Z}}_2)\) $H^{\ast }(\tilde{G}_{n,3}^{};{\mathbb{Z}}_2)$ of the Grassmann manifold \(\widetilde{G}_{n,3}\) $\tilde{G}_{n,3}^{}$ of oriented 3-dimensional subspaces of \({\mathbb {R}}^n.\) ${\mathbb{R}}^n.$ A complete description of \(H^*(\widetilde{G}_{n,3};{\mathbb {Z}}_2)\) $H^{\ast }(\tilde{G}_{n,3}^{};{\mathbb{Z}}_2)$ is given in the cases \(n=2^t-3\) $n=2^t-3$ and \(n=2^t-2,\) $n=2^t-2,$ while in the case \(n=2^t-1\) $n=2^t-1$ we obtain a description complete up to a coefficient from \({\mathbb {Z}}_2.\) ${\mathbb{Z}}_2.$